## Abstract

## Objectives

This study evaluated and compared Weibull parameters of resin bond strength values using six different general-purpose statistical software packages for two-parameter Weibull distribution.

## Methods

Two-hundred human teeth were randomly divided into 4 groups ( *n *= 50), prepared and bonded on dentin according to the manufacturers’ instructions using the following resin cements: (i) Variolink (VAN, conventional resin cement), (ii) Panavia21 (PAN, conventional resin cement), (iii) RelyX Unicem (RXU, self-adhesive resin cement) and (iv) G-Cem (GCM, self-adhesive resin cement). Subsequently, all specimens were stored in water for 24 h at 37 °C. Shear bond strength was measured and the data were analyzed using Anderson–Darling goodness-of-fit (MINITAB 16) and two-parameter Weibull statistics with the following statistical software packages: Excel 2011, SPSS 19, MINITAB 16, R 2.12.1, SAS 9.1.3. and STATA 11.2 ( *p *≤ 0.05). Additionally, the three-parameter Weibull was fitted using MNITAB 16.

## Results

Two-parameter Weibull calculated with MINITAB and STATA can be compared using an omnibus test and using 95% CI. In SAS only 95% CI were directly obtained from the output. R provided no estimates of 95% CI. In both SAS and R the global comparison of the characteristic bond strength among groups is provided by means of the Weibull regression. EXCEL and SPSS provided no default information about 95% CI and no significance test for the comparison of Weibull parameters among the groups. In summary, conventional resin cement VAN showed the highest Weibull modulus and characteristic bond strength.

## Significance

There are discrepancies in the Weibull statistics depending on the software package and the estimation method. The information content in the default output provided by the software packages differs to very high extent.

## 1

## Introduction

In recent years, the demand for non-metallic restorations in dentistry has increased, based on the enhanced esthetic outcome. Glass–ceramic fixed dental prosthesis (FDPs) must be adhesively cemented, and by using resin cements, the stability and the clinical long-term successes are improved . Resin cements not only improve the retention of dental restorations but also exhibit low solubility and microleakage in the oral environment compared with traditional cements, such as glass-ionomer or zinc phosphate cement . A multistep pretreatment is required to achieve a compound between the cement and the tooth. Therefore, a multistep adhesive system is needed to remove the smear-layer and additionally create a hybrid layer between the tooth and the cement . Adhesive systems show a high sensitivity for user handling errors which results in significantly lower bond strength values on dentin, based on the fact that the surface is contaminated with saliva or blood or when manufacturers recommendations are not fulfilled .

These challenges resulted into the development of self-adhesive resin cements, which represent a new system of adhesive bonding on teeth without the necessity of dentin pretreatment. The bonding on dentin without the removal of the smear-layer facilitates the clinical handling and enables the clinician to work more effectively. These resin cements combine the advantages of conventional resin cements, such as adhesive luting to the teeth, reinforcement of the reconstruction, and binding on all different kinds of framework materials, with the advantages of the passive luting agents (simple use without pretreatment of the dentin and less technical sensitivity).

The resin cements are only good, if they have simple application steps. Apart from the chemical composition, handling can impact the bond strength results, mainly the standard deviation and consequently the precision and quality of the adhesion. McCabe and Carrick suggested that less emphasis should be placed on the mean values and standard deviations of the strength when evaluating mechanical properties. Data of mechanical tests of materials may be analyzed by a Weibull distribution in which failure probability can be predicted at any level of stress, providing information about the variability of the results and reflecting the structural precision of the materials or bonded assemblies . Three- or two-parameter Weibull distributions can be considered. Harper et al. showed that the estimation of the three-parameter Weibull in MINITAB and SAS is possible. However, one should be cautious when using SAS. According to them SPSS and STATA offer just the two-parameter Weibull. Therefore, an example of the three-parameter Weibull estimation was computed exclusively with MINITAB. Anderson–Darling goodness-of-fit estimates were computed to find the best fitting distribution in each resin group. Typical analysis in dentistry considers sample sizes not exceeding 20 per group. According to Abernethy for small samples (at most 20 observations per group) estimation of the three-parameter Weibull is not creditable. Best practice suggested by him is to always use the two-parameter Weibull in such a case. The two-parameter Weibull distributions (with threshold set to 0) is supported by all well-established, general-purpose statistical software packages considered in this investigation, such as MINITAB, R, SAS, SPSS and STATA. When performing a statistical analysis the user generally expects the key statistical results to be the same from different software packages. However, these statistical software packages use different estimation methods for the parameters of the two-parameter Weibull distribution. Typically, the user is not explicitly informed which estimation technique is used by the statistical program. Therefore, the aim of this study was to calculate the two-parameter Weibull statistic with five different general-purpose statistical software packages and to compare the results and the information contained in their output.

The first hypothesis of this study was to test whether all used statistical software packages facilitating the calculation of the two-parameter Weibull statistic showed similar Weibull moduli and characteristic bond strength estimates. The second hypothesis was to test whether similar information content in the default output for the four tested resin cement groups was provided by the five statistical software packages. The third hypothesis was to test whether the four cement groups do not differ with respect to their modulus ( *m *) and characteristic bond strength ( *s *) of the underlying two-parameter Weibull distribution.

## 2

## Materials and methods

Two conventional resin cements with self-etch adhesive systems: Variolink II (VAN) and Panavia21 (PAN), and two self-adhesive resin cements namely, RelyX Unicem (RXU) and G-Cem (GCM) were tested in this study ( Table 1 ).

Abbreviations | Resin cement brands | Manufacturers | Batch-No | |
---|---|---|---|---|

Conventional cements | VAN | Variolink II Adhesive System: Syntac classic |
Ivoclar Vivadent, Schaan, Liechtenstein | K41833/K39878 J280035/J27820) |

PAN | Panavia21 Adhesive System: ED Primer A/B |
Kuraray Dental Co Ltd., Osaka, Japan | 408CA | |

Self-adhesive cements | RXU | RelyX Unicem | 3M ESPE, Seefeld, Germany | 361930 |

GCM | G-Cem | GC Europe, Leuven, Belgium | 0801091 |

## 2.1

## Specimen preparation

For this investigation, 200 caries-free human molars were collected and directly cleaned. Consequently, all teeth were stored in 0.5% chloramine T solution (Sigma–Aldrich, Seelze, Germany) at room temperature for a maximum of 7 days. Afterwards, they were stored in distilled water for a maximum of 180 days at 5 °C until the bond strength tests.

In order to achieve flat surfaces, the buccal surfaces of the teeth were leveled parallel to the tooth axis using a polishing machine (LaboPol-21, Struers, Bellerup, Denmark) with SIC paper P400 (ScanDia, Hagen, Germany). The teeth were then embedded in self-curing acrylic resin (ScandiQuick, ScanDia) and ground finished with carbide polishing paper (P500, ScanDia) until a sufficient bond area of 4 mm × 4 mm dentin surface was exposed.

The embedded teeth ( *N *= 200, *n *= 50 per cement) were then randomly divided into 4 cement groups (VAN, PAN, RXU, GCM). Prior to the application of the resin cement systems, the dentin test area was ground with the fine polishing paper P500 under water-cooling. While the self-adhesive cements did not require dentin conditioning, the corresponding adhesive systems of the conventional resin cements were applied to the dentin prior to cementation according to the manufacturer instructions. The specimen’s stepwise preparation is shown in Fig. 1 a–h . Subsequently, the resin cements were activated and placed in an acrylic cylinder (inner diameter: 2.9 mm) that was pressed onto the dentin surface by a holding device ( Fig. 1 a). In order to achieve homogeneous dispersion of the cement in the acrylic cylinder, a hexagonal steel screw with an outer diameter of 2.8 mm was inserted parallel to the long axis of the acrylic cylinder in its center ( Fig. 1 b). A load of 100 g was applied on the screw ( Fig. 1 c) and the excess cement was removed using microbrushes ( Fig. 1 d). This procedure allowed resin cement thickness in non-luted state of 0.5 mm in all specimens. VAN, RXU and GCM were dual-polymerized ( Fig. 1 e) and PAN was chemically polymerized.

## 2.2

## Shear bond strength test

The prepared bond strength specimens ( Fig. 1 f) were analyzed in the Universal Testing Machine (1 mm/min; Zwick/Roell Z010, Zwick, Ulm, Germany). The specimens were positioned in the sample holder of the testing machine with the tooth surface parallel to the chisel-shaped loading piston loading with a width of 1 mm ( Fig. 1 g) and then loaded until fracture ( Fig. 1 h). The bond strength (MPa) was calculated using the following formula: force to failure (N)/bonding area (mm ^{2 }).

## 2.3

## Statistical methods

The cumulative distribution function for the three-parameter Weibull distribution is defined by:

where *s *is called scale or characteristic value, *m *indicates the shape or Weibull modulus and *s *_{0 }denotes the threshold (location, minimum life, origin, guaranteed minimum life, shift). *G *( *x *) is usually interpreted as the probability of fracture for a test specimen and “ *x *” is the shear bond strength of the specimen tested. If *s *_{0 }is positive, it provides a guaranteed failure free period from 0 to *s *_{0 }. A non-zero threshold parameter should not be used unless it is anchored in the physics of the failure process. Frequently the two-parameter Weibull is considered for small sample sizes. It is obtained by assuming that the threshold is equal to zero ( *s *_{0 }= 0). In such a case the following probability density function:

is obtained. Frequently the Weibull statistic is computed based on the statistical approach .

Although the parameters of the underlying Weibull distribution are denoted by *s *and *m *their estimates obtained from the analysis of the data are identified by “hats” leading to ˆs

s ˆ

and ˆm

m ˆ

respectively. Note that at least five different parameterizations of the Weibull distribution have been proposed .

## 2.4

## Least squares estimates (LS)

The application of the LS approach to the parameters of the Weibull distribution is justified by the following property of the Weibull cumulative distribution function *G *( *x *) for *s *_{0 }= 0 with log denoting the logarithm with basis *e *:

In order to compute LS, in each sample group, ˆGi

G ˆ i

has to be the estimated for each observation . After ordering the data from the smallest to the largest value the *i *th position is considered to be a representative population percentage near to which the *i *th ordered observation falls. After taking the logarithm of the bond strength ( *X *-axis) it is plotted against log(−log(1−ˆGi))

log ( − log ( 1 − G ˆ i ) )

on the *Y *-axis and the linear regression fit of the scattered points is computed by means of LS with the sum of squares of the vertical distances being minimized. In other words log(−log(1−ˆGi))

log ( − log ( 1 − G ˆ i ) )

is regressed onto log( *x *). Due to the equation stated above, the estimate of the modulus ˆm

m ˆ

of the assumed underlying Weibull distribution is the slope of the linear regression. In order to obtain a value of ˆGi

G ˆ i

McCabe and Carrick suggested that the specimens within one test group should be ranked by calling the weakest specimen as “rank 1” and the strongest as “rank *n *”. The probability of failure ˆGi

G ˆ i

for each specimen from a group containing *n *specimens is given by

where *R *_{i }is the ranking number of the specimen. This way of computing ˆGi

G ˆ i

is frequently called mean rank and can be also found in MINITAB and SPSS under differing names, specifically, Herd–Johnson and Van der Waerden respectively.

Many alternative methods for estimating ˆGi

G ˆ i

in practical applications were suggested . In MINITAB software, several options such as median rank, modified Kaplan–Meier, and Kaplan–Meier (Hazen) are provided for ˆGi

G ˆ i

estimation. Conversely, in SPSS Rankit, Tukey and Blom choices are used. Abernethy recommends median rank: